Proving the right to bear a coat of arms is kind of like proving a line of descent for a genealogicial society. There's a whole lot of paperwork involved. Many arms could be inherited by a first born son, but there are a lot of details that make it more complicated than that. The College of Arms would certainly be the organization to know one way or the other.
It is important to remember that arms were awarded to an individual, not a family. Only one person has the right to bear a particular coat of arms at a time. There are many companies on the internet that sell artwork of supposed coats of arms based on last names, but that's not the way it really works.